A number that can be written without a fractional part is known as an integer (from the Latin word, which means “whole”). For example: integers are 21, 4, 0, and 2048, while 9.75, 5+1/2, and 2 are not. The positive natural numbers (1, 2, 3,…), also known as whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., -1, -2, -3,…) make up the collection of integers. The boldface (Z) or normal text Z — which stands for the German word ‘Zahlen’ — is often used to denote the set of integers. The set of all rational numbers Q, which is a subset of the real numbers R, is a subset of Z. Z is countably infinite, much like the natural numbers.
The integers are considered the smallest collection of natural numbers and the ring containing them. The integers are often referred to as rational integers in algebraic number theory to differentiate them from the more general algebraic integers. Rational integers, however, are algebraic integers that also fall under the category of general integers.
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Symbol of Integers:
The symbol Z is officially used by mathematicians to denote an integer. Different authors have used the symbol Z in different ways for mathematical expressions. Z, like natural numbers, is closed under addition and multiplication operations, meaning that the sum and product of any two integers is an integer. However, unlike natural numbers, Z is closed under subtraction when negative natural numbers are used. In the following sense, the integers form the most fundamental unital ring: for every unital ring, there is a special ring homomorphism from the integers into this ring. The ring Z is defined by its universal property of being an initial object in the category of rings. Since the quotient of two integers (e.g., 1 divided by 2) does not have to be an integer, Z is not closed under division. The integers are not closed under exponentiation, despite the fact that natural numbers are (since the result can be a fraction when the exponent is negative).
Different Properties Of Integers:
Closure Property:
When two integers are added or multiplied together, the product is an integer only, according to the closure property of integers. If a and b are both integers, the following is true:
- a + b = integer
- a x b = integer
Commutative Property:
If a and b are two integers, then, according to the commutative property of integers, then:
- a + b = b + a
- a x b = b x a
An important point of the commutative property is that it does not apply to operations of subtraction and division.
Associative Property:
If a, b, and c are all integers, then the associative property holds:
- a+(b+c) = (a+b)+c
- ax(bxc) = (axb)xc
Only addition and multiplication operations are affected by this property.
Distributive Property:
If a, b, and c are integers, then the distributive property of integers states that:
a x (b + c) = a x b + a x c
Additive Inverse Property:
If an is an integer, the additive inverse property of integers states that
a + (-a) = 0
As a result, the additive inverse of integer a is -a.
Multiplicative Inverse Property:
If an is an integer, the multiplicative inverse property of integers states that
a x (1/a)= 1
As a result, the multiplicative inverse of integer a is 1/a.
Identity Property:
Integers have the following identity elements:
- a+0 = a
- a x 1 = a
Applications In Computer Science:
In programming languages, an integer is often used as a primitive data form. However, since practical computers have a finite power, integer data types can only represent a subset of all integers. In addition, the intrinsic concept of sign in the common two’s complement representation distinguishes between “negative” and “non-negative” rather than “negative, positive, and 0.” (However, a machine can definitely decide whether or not an integer value is genuinely positive.) In some programming languages, fixed length integer approximation data types (or subsets) are denoted by an Integer (such as Algol68, C, Java, Delphi, etc.). Bignums and other variable-length representations of integers can store any integer that fits in the computer’s memory. Other integer data types have a fixed size, normally a power of two number of bits (4, 8, 16, etc.) or a recognisable number of decimal digits (e.g., 9 or 10).